Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries

Abstract

Mild to steep standing waves of the fundamental mode are generated in a narrow rectangular cylinder undergoing vertical oscillation with forcing frequencies of 3.15 Hz to 3.34 Hz. A precise, non-intrusive optical wave profile measurement system is used along with a wave probe to accurately quantify the spatial and temporal surface elevations. These standing waves are also simulated by a two-dimensional spectral Cauchy integral code. Experiments show that contact-line effects increase the viscous natural frequency and alter the neutral stability curves. Hence, as expected, the addition of the wetting agent Photo Flo significantly changes the stability curve and the hysteresis in the response diagram. Experimentally, we find strong modulations in the wave amplitude for some forcing frequencies higher than 3.30 Hz. Reducing contact-line effects by Photo-Flo addition suppresses these modulations. Perturbation analysis predicts that some of this modulation is caused by noise in the forcing signal through ‘sideband resonance’, i.e. the introduction of small sideband forcing can generate large modulations of the Faraday waves. The analysis is verified by our numerical simulations and physical experiments. Finally, we observe experimentally a new form of steep standing wave with a large symmetric double-peaked crest, while simulation of the same forcing condition results in a sharper crest than seen previously. Both standing wave forms appear at a finite wave steepness far smaller than the maximum steepness for the classical standing wave and a surface tension far smaller than that for a Wilton ripple. In both physical and numerical experiments, a stronger second harmonic (in time) and temporal asymmetry in the wave forms suggest a 1:2 resonance due to a non-conventional quartet interaction. Increasing wave steepness leads to a new form of breaking standing waves in physical experiments.

Surface-wave damping in a brimful circular cylinder

“…Let us point out here that the difficulty appears also for other shapes of the container (e.g. for rectangular containers, see Jiang et al 1996) and for related fluid configurations such as liquid bridges, where the difficulty was solved by Higuera, Nicolás & Vega (1994) (see also Higuera & Nicolás 1997).…”

Sloshing motions in excited tanks

“…In particular, the vertical tank excitation, as originally explored in FaradayÕs experiment [14], has had much attention (e.g. the review papers by Miles and Henderson [30]; Perlin and Schultz [32] and the recent work by Jiang et al [24,25]). The literature also reveals extensive research on pure horizontal excitation, in general, especially in relation to the application of idealised seismic events [8] and ship stability [10].…”

Faraday instability threshold in large-aspect-ratio containers

“…Among the still unresolved questions, linear damping is not completely understood for low viscosity at modérate aspect ratio, even if the effects of contact line dynamics and surface contamination are eliminated (Henderson & Miles 1994;Martel, Nicolás & Vega 1998;Howell et al 2000). The theoretical and experimental determination of the instability threshold has received considerable attention, both in the modérate Jiang et al 1996) and large (Douady 1990;Edwards & Fauve 1994;Kumar & Tuckerman 1994;Bechhoefer et al 1995;Christiansen, Alstrom & Levinsen 1995;Kumar 1996;Lioubashevski, Fineberg & Tuckerman 1997) aspect-ratio limits. In particular, if lateral walls are ignored the instability threshold is reliably calculated for arbitrary viscosity by a numerically cheap method (Kumar & Tuckerman 1994).…”